Example Question - factoring denominator

Limit Calculation of Rational Function

The limit given in the image is \[ \lim_{{x \to 5}} \frac{x - 5}{x^2 - 25} \] We can start by factoring the denominator: \[ x^2 - 25 = (x - 5)(x + 5) \] The expression for the limit then becomes: \[ \lim_{{x \to 5}} \frac{x - 5}{(x - 5)(x + 5)} \] We can simplify the expression by canceling out the common factor of \(x - 5\) in the numerator and the denominator: \[ \lim_{{x \to 5}} \frac{1}{x + 5} \] Now we can directly substitute \(x = 5\) into the simplified expression, as there are no more discontinuities: \[ \frac{1}{5 + 5} = \frac{1}{10} \] So, the value of the limit is \(\frac{1}{10}\).